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Fractional calculus was introduced in one of Niels Henrik Abel's early papers [3] where all the elements can be found: the idea of fractional-order integration and differentiation, the mutually inverse relationship between them, the understanding that fractional-order differentiation and integration can be considered as the same generalized ...
In mathematics, the Grünwald–Letnikov derivative is a basic extension of the derivative in fractional calculus that allows one to take the derivative a non-integer number of times. It was introduced by Anton Karl Grünwald (1838–1920) from Prague , in 1867, and by Aleksey Vasilievich Letnikov (1837–1888) in Moscow in 1868.
In fractional calculus, these formulae can be used to construct a differintegral, allowing one to differentiate or integrate a fractional number of times. Differentiating a fractional number of times can be accomplished by fractional integration, then differentiating the result.
Working with a properly initialized differ integral is the subject of initialized fractional calculus. If the differ integral is initialized properly, then the hoped-for composition law holds. The problem is that in differentiation, information is lost, as with C in the first equation.
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Igor Podlubny's collection of related books, articles, links, software, etc. Podlubny, I. (2002). "Geometric and physical interpretation of fractional integration and fractional differentiation" (PDF). Fractional Calculus and Applied Analysis. 5 (4): 367– 386. arXiv: math.CA/0110241. Bibcode:2001math.....10241P. Archived from the original ...
The Coopmans approximation is a method for approximating a fractional-order integrator in a continuous process with constant space complexity.The most correct and accurate methods for calculating the fractional integral require a record of all previous history, and therefore would require a linear space complexity solution O(n), where n is the number of samples measured for the complete history.